The only time you see a bouncing logo is when your TV is restarting or powering up from being unplugged. You should never be seeing a bouncing logo unless you removed the power physically from your Roku or you restarted it.
That is true if everything is working properly. In this case the bouncing logo is appearing when it should not. I am not sure if this is a Roku problem or an LG problem. You may have already addressed this and I missed it but have you contacted LG? It almost sounds like somehow the TV is cutting power to the Roku operating system when you turn off the TV. If you turn off the TV and wait about three minutes to turn it back on do you get the bouncing logo? If not that could mean the TV is actually turning off after a certain time and not just going to sleep. I am stumped at this point.